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Z=61 Task 3: Answer the following: a. Evaluate: Siz cos(x) sin?(x) dx (10 Marks) b. The...
value of z= 96Task 3: Answer the following:a. Evaluate: \(\int_{\frac{\pi}{2}}^{\pi} \boldsymbol{Z} \cos ^{3}(x) \sin ^{2}(x) d x\)b. The moment of inertia, \(I\), of \(a\) rod of mass ' \(m^{\prime}\) and length \(4 r\) is given by \(I=\int_{0}^{4 r}\left(\frac{Z m x^{2}}{2 r}\right) d x\) where \(^{\prime} x^{\prime}\) is the distance from an axis of rotation. Find \(I \)Task 4: Answer the following:Using the Trapezoidal rule, find the approximate the area bounded by the curve\(y=\boldsymbol{Z} e^{\left(\frac{x}{2}\right)}\), the \(\mathrm{x}\) -axis and coordinates \(x=0,...
1. Numerical Integration The integral of a function f(x) for a s x S b can be interpreted as the area between the f(x) curve and the x axis, bounded by the limits x- a and x b. If we denote this area by A, then we can write A as A-f(x)dx A sophisticated method to find the area under a curve is to split the area into trapezoidal elements. Each trapezoid is called a panel. 1.2 0.2 1.2 13...
Question 8 a) Find: (3x3 – 5x2 + x - 4) dx (4 marks) b) Use the trapezium rule with 6 strips to approximate the area bounded by the curve y = (x - 1, the x -axis and the lines x = 2 and x = 5 (6 marks)
4. Solve the following equation using integration by parts. (10 Marks) | 22 x2 In(x) dx 5. Calculate the shaded area between the curve and the x-axis as shown below. (5 Marks) y = x2 - 6x
ENG 1005 ASSIGNMENTI QUESTIONS (1) Use integration by parts to calculate sin(In(x) dx and Here, In is the natural logarithm. cos(In(x))dx. [5 marks (2) (a) Use integration by parts on sinh(t) sinh(t)dt and the identity cosh (1) = 1+sinh'in to calculate the integral of sinh(r). (b) Calculate the integral of sinh(r) by expanding the product and then integrating, Confirm that you get the same answer as in part (a). (e) Show that if x is a positive real number, then...
Question 2 (Learning Outcome 2) 0 S (*x+3) dx S A) Evaluate the following integrals. 4x+7 2x+5) 5x2–2x+3 (ii) dx (x2+1)(x-1) x2+x+2 (iii) S3x3 –x2+3x+1 dx dx (x+1)V-x-2x In (x) dx (iv) S x2 X+1 (vi) S dx (1+x2) (vii) S dx x(x+Inx) (viii) Stancos x) dx (ix) 30 Sin3 e*(1 + e*)1/2 dx dx 2 sin x cos x (x) S B) Find the length of an arc of the curve y =*+ *from x = 1 to x...
6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch the region. 5x2-2. f(x)--3x2 + 30 and g(x) marks) c) Evaluate the following integrals. Give your answers to 2 decimal places. dac i) 2 (3+2 dx. (8 marks) [25 marks] 6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch...
2. Evaluate the following indefinite integrals: (a) vel V=(x+2) dx ET (b) 3. Evaluate the following definite integrals: (a) cos(x) da (sin(x) +18 (b) COS 4. The graph of y=g(t) is shown below, and consists of semicircles and line segments. y=g() -1 3 6 596 s(t) dt Define the function f(x) by f(x)= Use the graph of y = g(t) and the properties of the definite integral to find: (a) the value of (i) f(3) (ii) f(-1) (iii) 1'(6) (b)...